Final answer:
To find the vector function that represents the curve of intersection between the paraboloid and the cylinder, equate the equations of the paraboloid and the cylinder. Let t be the parameter, and the vector function can be written as r(t) = t, t, 5t².
Step-by-step explanation:
To find a vector function that represents the curve of intersection between the paraboloid and the cylinder, we need to equate the equations of the paraboloid and the cylinder.
The equation of the paraboloid is z = 8x² - 3y² and the equation of the cylinder is x² + y² = r², where r is the radius of the cylinder.
Substituting the equations into each other, we get 8x² - 3y² = r². To find the vector function, we can choose a parameter t and write the equations of x, y, and z as functions of t.
Let's choose t as the parameter. We can write x = t, y = t, and z = 8t² - 3t² = 5t².
Therefore, the vector function that represents the curve of intersection is r(t) = t, t, 5t².